KMS states, topological entropy and the variational principle for full periodic C*-dynamical systems

To any periodic and full $C^*$--dynamical system $(\A, \alpha, {\Bbb R})$, an invertible operator $s$ acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of $s$. A Perron--Frobenius type theorem asserts the existence of KMS states at inverse temperatures equal the logarithms of the inner and outer spectral radii of $s$ (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self--similar sets in fractal geometry and noncommutative metric spaces are discussed. Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given. Unital completely positive maps $\sigma_{\{x_j\}}$ implemented by partitions of unity $\{x_j\}$ of grade $1$ are considered, resembling the `canonical endomorphism' of the Cuntz algebras. The relationship between the Voiculescu topological entropy of $\sigma_{\{x_j\}}$ and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case $\text{ht}(\sigma_{\{x_j\}})$ is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary. The measure--theoretic entropy of $\sigma_{\{x_j\}}$, in the sense of Connes--Narnhofer--Thirring, is compared to the classical measure--theoretic entropy of the subshift. A noncommutative analogue of the classical variational principle for the entropy is obtained for the `canonical endomorphism' of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of Cuntz--Krieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done.